Much of the recent work by the author [1, 2, 3, 4 and 5] as well as others [6, 7, 8, 9, 10, 11 and 12] has focused on the need to consider idealization geometry a variable in the finite element solution procedure. Such solutions, in which grid geometry is included as a solution variable, have become known as optimum grid solutions.
To date, considerable emphasis has been placed on establishing criteria for the selection of optimum grid solutions. Criteria have been successfully established for the elastostatic, elastodynamic and the elastic stability problem [5]. Focus is now being directed at exploring the benefits of utilizing optimum grid solutions. Several papers have communicated techniques for determining “near optimum” meshes [4,12].
The purpose of this paper is to consolidate and present the recent advanced made in the area of optimum grid solutions.
The author has arranged this talk into three subareas that serve as objectives:
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(1)
To present the user of the finite element method with a basic understanding of the theoretical foundation of the optimum grid solution. The formulation of the elastostatic problem will be reviewed. A method for formulating the elastodynamic and elastic stability problem will be indicated. The physical characteristics indicative of the optimum grid solution for the later class of problems will be indicated.
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(2)
To present the user with a compendium of user oriented guidelines that have been presented in the published literature. As mentioned, methods for generating “near optimum” solutions have recently received attention. The purpose of this section will be to perform a comparative evaluation of criteria used to direct mesh refinement.
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(3)
To indicate to the user the benefits and pitfalls (cost) of full grid optimization versus the near optimum solution. Experience the author has had in operating a program capable of either “mode” shall serve as the basis of this section.
Throughout this talk simple example problems will be presented that demonstrate the nature of the process being discussed.
Guidelines regarding convergence, error analysis, and error detection as they are relevant in the optimum grid context are included.